In the previous section, we saw that it is possible to get behavior of considerable complexity just by applying a variety of operations based on simple arithmetic. In this section what I will show is that with the appropriate setup just addition and subtraction turn out to be in a sense the only operations that one needs.
The basic idea is to consider a sequence of numbers in which there is a definite rule for getting the next number in the sequence from previous ones. It is convenient to refer to the first number in each sequence as f, the second as f, and so on, so that the nth number is denoted f[n]. And with this notation, what the rule does is to specify how f[n] should be calculated from previous numbers in the sequence.
In the simplest cases, f[n] depends only on the number immediately before it in the sequence, denoted f[n–1]. But it is also possible to set up rules in which f[n] depends not only on f[n–1], but also on f[n–2], as well as on numbers still earlier in the sequence.
The table below gives results obtained with a few specific rules. In all the cases shown, these results are quite simple, consisting of sequences that increase uniformly or fluctuate in a purely repetitive way.
Examples of some simple recursive sequences. The nth element in each sequence is denoted f[n], and the rule specifies how this element is determined from previous ones. With all the rules shown here, successive elements either increase smoothly or fluctuate in a purely repetitive way. Sequence (c) is the powers of two; (d) is the so-called Fibonacci sequence, related to powers of the golden ratio (1+Sqrt)/2≃1.618. All rules of the kind shown here lead to sequences where f[n] can be expressed in terms of a simple sum of powers of the form an.